I’ve been posting a bit on the Cambridge Coaching website recently (they’re a tutoring service where I’ve been working). My writing there has been mostly for students and parents, rather than being geared toward other educators, like this one.
Four Mathematicians You Should Know – quick bios of famous mathematicians and how they tell a different story about what math is than what kids sometimes learn in school
How to Help Your Child with Math Homework: 5 Easy Questions You Can Ask – geared toward parents who don’t feel comfortable with math; I explain how important they are to supporting their kids and how valuable they can be
This semester, for the first time in almost a decade, I’m taking a math class with people who are not involved with math education. It’s really bringing me back.
Working in my classroom, I found that having students work in groups was one of the most powerful tools of instruction that I had. This view is supported by research and by experiences I have had, since college, as a learner. There’s nothing that can replace students working through new ideas together — a teacher who already knows the content runs the risk of giving too much away, whereas students working together will (usually) be able to give each other new ideas or questions that provide each other with just the right amount of new information. Even when this doesn’t work perfectly, students get to watch mathematics in action as somebody similar to them in content knowledge works through new ideas.
So, I’ve been living in that bubble. Pretty much everyone I talk to about teaching agrees that group work is an important aspect of mathematics pedagogy, and that math is a collaborative pursuit. Then I go to class one day and hear the graduate student filling in for my professor say, “If you don’t understand this, work it out alone later. Mathematics is something you do by yourself.” (I’m paraphrasing.) He repeated a similar sentiment in the following lecture.
A few days later, I was chatting with a classmate before the professor began, and he spoke about how he liked mathematics so much because it was a solitary pursuit. He said he probably would not be interested in the subject otherwise.
Oh right. I remember now why I hated math class so so so much.
We’re missing out on a lot of great mathematicians. Folks see many people working in math who enjoy working on their own, so they assume that there is a connection between liking math and liking working alone. But! Maybe we see people like that in mathematics because math is taught as an independent pursuit, so most of the people who persevere are those who have an innate preference toward working that way.
Sure, sometimes you need to sit and ponder something on your own, but there’s no reason that math should be so incredibly solitary! If math class were to incorporate more collaborative pedagogy, students who enjoy working with others may be more likely to find success in the field and therefore persevere — if only for the simple fact that they enjoy it more. We may discover strong math ability in people who would traditionally be turned off by the way they see mathematics traditionally being done. Then, beautifully, we have a more diverse group of role models for the next generation.
I won’t even get into the gender stuff this brings up. (Well, maybe just a teaser: young girls are taught how important it is that they orient themselves toward other people and be nice and helpful; how can we expect very many women to break down that expectation or the popular view of math as a solitary endeavor?)
As I was beginning to collect resources about creativity and how it is understood in the context of mathematics education earlier this month, I was surprised to find a large number of articles focusing on the instruction of ‘gifted and talented’ students.
Since I began teaching, I have had no real interest in students identified in this way. They’re somewhat less interesting to teach, quite frankly, because they’re most likely going to learn the content no matter what you teach. Most of students have a great deal of support at home, which has allowed their intellect and curiosity to be displayed to their full extent. Many of these students also are identified in this way also just because they happen to learn in a way that schools typically expects students to learn. As a teacher, you don’t have to do much in the way of reflecting upon and adjusting your practice in order to get these students to reach course curriculum goals.
I learned a lot more about teaching from my ‘level one’ classes than my ‘honors’ or AP groups. These students forced me to think about how to present the essence of mathematics that makes it so captivating.
Seeing the ‘gifted’ students being called out in particular as being creative made me think more about the difference between the students who are viewed as ‘good at math’ and those who are not.
Is creativity the thing that makes people think certain students are gifted? If so, doesn’t that imply that creativity is the sign of a good mathematician, which should make it the goal of math instruction? And yet, we focus on teaching mostly skills to the students who are already succeeding, rather than helping them to become creative in mathematics. This means we are giving up on these students ever being truly proficient in mathematics.
Or, maybe the implication is slightly different — creativity isn’t the thing that causes folks to identify a student as gifted; ability to answer math questions of some sort is. But, the amount of studies on gifted students and creativity still demonstrates that we see more creativity in the gifted students. This tells me that creativity is a tool that helps these students to succeed in their math classes. Again, why are we not finding a way to give this tool to all students?
Overall, it seems to me that if researchers and educators note overlap in the students they consider to be ‘creative’ and the students they consider to be ‘gifted,’ then not making creativity an important part of math education for all students is hugely problematic. The issue further complicated by the economic, gendered, and racial differences between students who are identified as ‘gifted’ and those who are not.
Since my second year teaching, I’ve been advising clubs. Some have gone away, and some have grown to be very popular and effective. It’s one of my favorite parts of my job, because you get to see students voluntarily give time and energy to something that is important to them. I have gotten to know some of my students very well through different clubs, which sometimes even wraps around to help us to do well together in math class. I have also learned a ton about managing goal-based organizations, which has been very useful to me outside of school.
Obviously, it is best if the students accomplish as much as possible on their own. When the students are the ones making change, they are doing things that speak most truly to their and to other students’ needs. They also get the opportunity to learn a lot of very useful skills as they plan and execute fundraisers, events, or informational campaigns.
On the other hand, kids are kids. They haven’t lived for very long yet, so sometimes they’re not very good at things. I know that in some cases (but not all!), if I just do something myself, it will get done so much more efficiently. Sometimes this is important, if the goal is something that the wider school community could benefit from.
My goal is always that my students accomplish as much as possible, which sometimes requires me to lend my experience. A certain amount of failure and floundering is important for learning, but I don’t expect my students to always come up with effective tools from scratch. My M.O. is to share my knowledge, but to have the students be the ones to actually do as many things as possible, including planning, goal-setting, and making decisions.
Here are some guidelines that I have developed over the past few years:
- Elect leaders. Make sure they have well-defined responsibilities.
- Check in with the leaders from time to time to remind them of their responsibilities and to share tips about managing the other club members.
- Ensure that your group has a common goal or interest.
- Set up protocols and present them to your students. In the GSA that I co-advise, the grown-ups run the first few meetings of the year. We do so with the same simple template each time, which includes a check-in and having an agenda on the board. Now our president runs his meetings in the same way.
- Unless you feel absolutely certain that students will do this quickly on their own (note that I said *will* and not *can*), set up a simple way for club members to communicate. My students all have google accounts through the school, so I have found google groups to work nicely.
- When tasks are being discussed, make sure particular students take responsibility for particular tasks with specific deadlines.
- If you have a group without strong student leadership, come to conclusions by leading group brainstorms. I find that the think-pair-share model works very nicely in clubs, especially if you ask students to physically write things down. As a teacher, I will often add my own ideas to a brainstorm, but make sure that they don’t carry more weight than any other ideas.
- Don’t run a club if no students want it to exist. (There are definitely exceptions. Even if there were no students that wanted to have a Gay-Straight Alliance, I would probably still want to force it into existence.)
I’ll probably go into detail about a few particular things that the students in the clubs that I work with have accomplished in future posts, because they do oftentimes wow me. The GSA just raised $200 for a homeless shelter for youth, the Feminist Coalition is putting together a zine, and the student director for the talent show was the only reason that we grown-ups were able to relax and enjoy the show.
Does anybody have any other tips about empowering students to take charge and make change in school-based clubs?
Hey it has been a million years! I have kind of just been doing a lot of the same stuff with my algebra 1 and precalculus classes (with a few exceptions… those should appear on the blog in a bit!). I have been teaching one new class this year, but haven’t felt like I had taken enough ownership of anything to share it here.
The class an introductory Computer Science is run by TEALS which means the projects and plans are theirs. It’s a cool class. We program in Snap!, a visual block-based language. I have a few outside programming professionals who come in to help, which is good. The material is all pretty darn simple, but I don’t have the kind of scope that tells me what people will really use when they work as programmers, in the way that I can do with math.
Anyway, for our last unit, we’re having our students design their own game. Rather than teaching anything like loops or lists or informative things like that, this unit is about more general concepts like good game design, project management, and working in a team. Since the unit is right after break, I had some time to really get into it and I’m pretty excited about what I’ve come up with. The first few days talk about some background information and allow for a bit of brainstorming before diving headlong into making a project.
After these things, I’ll give them a few days to code. Then we’ll spend a day each on debugging, testing prototypes, and the life of a video game developer. I’ll post those lessons later (aka, after I make them, as they do not yet exist).
Here’s the stuff! Use it if it is useful to you!
Day 1 – Intro to game design
- powerpoint: what makes a good game?
- Student activity
Day 2 – Intro to team programming
- powerpoint: working in a team, pair programming (includes practice)
- Student activity
- make a list of students’ strengths
- review work from day before and establish who will do what
Day 3 – History of games
- powerpoint: important games
- Student activity
- what do they like about those games? dislike?
- decide whether they would like to incorporate any of those elements into their game
- if time, let them look through the links of video game timelines in the resources
Day 4 – Planning Protocols
- powerpoint: sharing information, flowcharts, pseudocode
- Student activity
- create sharing resources (shared google drive folder, shared snap account)
- handout: fill out project planning document *they should access + update this online
Awhile back, I wrote about the portfolios that I was having my students create over the course of the school year. I have now run this project for two years, and wanted to describe more details about how I set up this experience for my students, as well as talk a little bit about how I know that it is valuable for them, and changes that I plan to make for next year.
The portfolio that my students create is a little booklet. I leave the presentation entirely up to them, besides requiring that they submit it in person rather than electronically (just because I’m sick of kids claiming to have had email or google drive issues). Each portfolio consists of a cover page, a one to two page reflection on problem-solving, and written explanations of the four problems that the student solved that year.
I’d also like to note that a lot of the ideas described here come from Bryan Meyer’s blog Doing Mathematics and Jim Mahoney’s book, Power and Portfolios. Obviously I’ve tweaked everything so that it makes more suits my practice best, but I’ve definitely borrowed very heavily from them.
BEGINNING OF THE YEAR
There are a few important things that I do at the beginning of the year to set up the expectation that my students will be creating there own portfolios.
- On the first day of school, we spend most of the period on this activity that I’ve written about before. I give them an open-ended problem and provide them with very little information. They are forced to make decisions within the problem. These are manageable decisions, such as estimating how many students go to our school. It is also important that they need to write out all these decisions and the reasoning behind them, which is what they will be expected to do for the final portfolio.
- We read through the mathematicians’ habits using this handout.
- We set up our binders. There are instructions for how to do this on the handout mentioned in #2 and I have lots of examples from previous students.
- I show them portfolios written in past years.
Here are the habits I use:
Collaborate, Be confident/patient/persistent, Create data, Generalize, Look for patterns, Multiple representations, Seek why and prove, Solve a simpler problem
Students select a problem they solved in the last two weeks that is an example of them using one of the habits. They fill out a little cover sheet (I keep them in labeled pockets on my back wall) and put both the problem and the cover sheet in their binders. It’s super important that they actually submit the problem, so they can potentially write about them later.
Here‘s an example of one of the cover sheets. The others are nearly identical, with the name of the habit changed.
Each quarter, students select the problem of which they are the most proud and write a detailed account of how they figured it out. We call these ‘portfolio write-ups.’ At this time, I do not include any discussion of the mathematician’s habits in this assignment (see below for future changes I plan to make).
One issue is that sometimes the write-ups look more like how-to’s than I would like. It helps to encourage students to include things that they did that were incorrect, or times that they felt confused or overwhelmed. This also reminds students that you can still be a good mathematician even if you don’t understand something immediately.
I give them this handout, which contains instructions and a grading rubric.
In addition to their second-quarter portfolio write-ups, I also ask students to answer questions about the collection of problems in their binders and about how they are doing with the habits. I use this handout.
END OF YEAR
By the end of the year, my students have written 3 write-ups. Their final portfolio must have four, so they have one more to write. Of course, many students have more than one more to write, if they missed assignments earlier in the year or aren’t super well-organized.
Every quarter, I give fairly detailed feedback to the portfolio write-ups, so I do ask students to rewrite their work, unless they got a 100%. I don’t require this, though maybe I should.
This is the handout that I give to students.
This is the grading sheet that I use for each portfolio.
My only requirements for the presentation is that it has a cover sheet and that students hand in a physical copy. I give full points for presentation as long as they are neat and legible – I don’t care whether they simply typed “Meghan Riling’s Portfolio” in black ink, or created an intricate design in colored pencil. I recommend students type the write-ups, but have given full credit to immaculately handwritten ones.
HOW DO I KNOW IT’S WORKING?
First of all, what do I mean by ‘working’? Ideally it will do a few things for students: develop their problem-solving skills, make them believe in themselves as problem-solvers, make them believe that being good at math isn’t the same thing as being fast at math, and encourage them to try new things instead of give up when they are confused.
For one thing, my students take this project pretty seriously. For the most part, their write-ups go into a lot of depth. (I’m considering a length cap next year, as some honors kids handed in 15+ page portfolios.)
I also gather from what they write in their reflection letters that they do feel that their problem-solving skills are improving. It’s possible that they’re just sucking up to me. But I don’t think so — some kids were talking to me about how they thought they had pulled one over on me. In the reflection questions, I asked how they would use what they had learned in the future, and these dudes thought they were just so slick for saying they would use it in real life, rather than in math class. So, I guess they don’t even know that they’re saying things that would count as sucking up to me.
Here are some quotes I grabbed from their write-ups that give me hope:
Senior – precalculus
This year I learned I am actually very capable of figuring out Math. I used to be hard on myself when I couldn’t get the right answer and give up right then. That isn’t the case this year; I have learned there is possibility.
Junior – precalculus
I have learned not to shy away from things because failure is normal. I will fail numerous times in life but I need to bounce back and shake it off. If you set your mind to getting the problem right, you’ll get it right. … If I don’t get it right the first time, why can’t I get it right the second time?
Sophomore – honors algebra 2
What I learned about solving problems is that there is no correct way to solve a problem and there are almost always multiple ways to look at it.
Junior – precalculus
I have become much more creative as well as talkative and collaborative about how to solve the problems in front of me. I believe that the most useful skill I have learned was to really experiment with all different sorts of ways.
Freshman – algebra 1
See now one of my biggest problems when it comes to math is giving up easily. I used to get overwhelmed and anxious when I saw a difficult equation, which would result in me not completing a lot of problems. So by picking up the habit of being confident, patient, and persistent I’ve been able to be open minded to lots of new possibilities and outcomes.
I know that’s a bunch of them. The things that pop out a lot are that they now see there are multiple ways to solve a problem and that they are finding that they are solving more problems just by not giving up. I think the other side to not giving up is that they must be trying new approaches. And frankly, even if they are sucking up to me, I’m not that worried about it — at least they know that talking about perseverance and creative problem-solving are ways to suck up to me!
Anyway, I think that just about explains everything. I’m more than happy to explain anything at all in more detail, if anybody is interested.
After doing some programming in my Honors Algebra 2 class, I turned over to my freshmen Algebra 1 class. Very different group of kids, so I created a whole new intro. This time, I walked them through every tiny thing and didn’t expect them to do nearly as much from scratch.
The set-up that I used for the first two lessons was:
-kids coded in http://www.coffeescript.org
Then for the third, which only a handful got to, I used:
-a written hand-out with explanations only
-coding in sublime and running programs in firefox
The first two days went really well. Then on the third day, kids who hadn’t gone very quickly and were still on earlier lessons started to flag. I wasn’t really able to help all 22 kids in my big class as much as they needed, what with having to constantly add dropbox to various computers that it had mysteriously disappeared from, and the momentum had run out, so on the third day I saw much less enthusiasm. I really should have just moved them all onto the day 3 activity no matter what, because it was visual, which always helps, and since it was just a 3 day thing, it wasn’t crucial that they all did each lesson straight through.
A consistent problem was getting students to understand WHERE in my prepared code they were supposed to add their own things. I did absolutely no direct instruction, which may have helped with this. Then again, since we don’t have programming classes beyond the club that I run with Adam (aka an actual pro programmer), I thought it was nice to show them that they could just sit down at a computer without anybody telling them what to do, since that would most likely most clearly mirror their experience for the next few years if they chose to continue.
One very nice thing about this all is that one of my students who has recently had very low math self esteem, to the point of handing in blank tests and quizzes, was a total rock star and was even helping other kids get started with the third lesson.